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To solve the problem we propose a local-search algorithm. Iterative improvement within our algorithm gives rise to non-trivial optimization problems, which, for the measures of set intersection and Jaccard, we solve using a greedy method and non-negative least squares, respectively.

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- 1. overlapping correlation clustering francesco bonchi aris gionis antti ukkonen yahoo! research barcelonaMonday, September 26, 2011
- 2. overlapping clusters are very natural - social networks - proteins - documents 2Monday, September 26, 2011
- 3. most clustering algorithms produce disjoint partitions 3Monday, September 26, 2011
- 4. overlapping is conceptually challenging to formulate - why assign a point to a further center? - why/how to generate less good clusters? 4Monday, September 26, 2011
- 5. correlation clustering Ccc () = |s(u, v) − I((u) = (v))| 5Monday, September 26, 2011
- 6. 6Monday, September 26, 2011
- 7. 1 7Monday, September 26, 2011
- 8. 0 8Monday, September 26, 2011
- 9. 0.33 ??? 9Monday, September 26, 2011
- 10. 0.33 10Monday, September 26, 2011
- 11. 0.33 multiple labels = multi-cluster assignment 10Monday, September 26, 2011
- 12. 0.5 11Monday, September 26, 2011
- 13. 0.67 12Monday, September 26, 2011
- 14. 54167/108301 ??? 13Monday, September 26, 2011
- 15. overlapping correlation clustering Cocc () = |s(u, v) − H((u), (v))| 14Monday, September 26, 2011
- 16. comparing sets of labels H((u), (v)) - Jaccard coefﬁcient - set intersection indicator 15Monday, September 26, 2011
- 17. overlapping correlation clustering correlation clustering 16Monday, September 26, 2011
- 18. overlapping correlation clustering correlation clustering set of labels L, |L| = k 16Monday, September 26, 2011
- 19. overlapping correlation clustering correlation clustering set of labels L, |L| = k (u) ∈ L (u) ⊆ L 16Monday, September 26, 2011
- 20. overlapping correlation clustering correlation clustering set of labels L, |L| = k (u) ∈ L (u) ⊆ L C() = |s(u, v) − H((u), (v))| 16Monday, September 26, 2011
- 21. overlapping correlation clustering correlation clustering set of labels L, |L| = k (u) ∈ L (u) ⊆ L C() = |s(u, v) − H((u), (v))| |(u)| ≤ p 16Monday, September 26, 2011
- 22. dimensionality reduction - mapping to sets instead of vectors 17Monday, September 26, 2011
- 23. 18Monday, September 26, 2011
- 24. 18Monday, September 26, 2011
- 25. 18Monday, September 26, 2011
- 26. 18Monday, September 26, 2011
- 27. 18Monday, September 26, 2011
- 28. 18Monday, September 26, 2011
- 29. v u y x 19Monday, September 26, 2011
- 30. v u y x (u,v) (x,y) 19Monday, September 26, 2011
- 31. u∈V {v} expresses the error incurred by vertex v when it has the labels Now, giv (v), and the remaining nodes are labeled according to . The J(X, Sj subscript p in Cv,p serves to remind us that the set (v) should have at most p labels. Our general local-search strategy is summarized in Algorithm 1. Algorithm 1 LocalSearch which is 1: initialize to a valid labeling; 2: while Cocc (V, ) decreases do 3: for each v ∈ V do and we 4: ﬁnd the label set L that minimizes Cv,p (L | ); We obse 5: update so that (v) = L; to the un 6: return xi and t Equation Line 4 is the step in which LocalSearch seeks to ﬁnd an propose optimal set of labels for an object v by solving Equation (3). constrain This is also the place that our framework differentiates be- squares tween the measures of Jaccard coefﬁcient and set-intersection. optimiza B. Local step for Jaccard coefﬁcient variables Problem 3 (JACCARD - TRIANGULATION): Consider the set20 The s {S , z }Monday, September 26, 2011 , where S are subsets of a ground set U = drawbac
- 32. Jaccard triangulation given {Sj , zj }j=1...n ﬁnd X ⊆ U to minimize n d(X, {Sj , zj }j=1...n ) = |J(X, Sj ) − zj | j=1 21Monday, September 26, 2011
- 33. set-intersection indicator hit-n-miss sets √ O( n log n) approximation greedy approach 22Monday, September 26, 2011
- 34. experimental evaluation 23Monday, September 26, 2011
- 35. EMOTION: 593 objects, 6 labels YEAST: 2417 objects, 14 labels 24Monday, September 26, 2011
- 36. EMOTION YEAST 0.2 1 0.2 1 Precision Recall Precision Recall Cost/edge Cost/edge cost cost 0.1 prec 0.8 0.1 prec 0.9 rec rec 0 0.6 0 0.8 2 4 6 8 10 5 10 15 20 k k EMOTION YEAST F 0.1 1 1 r Precision Recall Precision Recall I Cost/edge Cost/edge cost 0.1 cost prec 0.5 prec 0.9 rec rec c p d 0 0 0 0.8 2 p 4 6 2 4 6 p 8 10 12 14 s t Fig. 1. Cost per edge, precision and recall of OCC-JACC as a function of 25Monday, September 26, 2011
- 37. EMOTION YEAST 0.05 0.6 0.04 1 Precision Recall Precision Recall 0.4 Cost/edge Cost/edge cost cost prec 0.03 prec rec 1 rec 0.8 0 0.02 0.8 0.01 0.02 0.03 0.04 0.01 0.02 0.03 0.04 q q Fig. 3. Pruning experiment using OCC-JACC. Cost/edge and precision and recall as a function of the pruning threshold q. In fact, the results for YEAST shown in ﬁgures 1 and 2 were 26 computed with q = 0.05. In terms of computational speedupMonday, September 26, 2011
- 38. protein clustering - pairwise similarities based on matching of amino-acid sequences - compare using a hand-made taxonomy 27Monday, September 26, 2011
- 39. TABLE II Precision, recall, and their harmonic mean F-score, for non-overlapping C non-overlapping clusterings of protein sequence datasets computed using SCPS [14] and the OCC algorithms. BL is the precision of a baseline that assigns all sequences to the same cluster. BL SCPS OCC-ISECT OCC-JACC dataset prec prec/recall/F-score prec/recall/F-score prec/recall/F-score D1 0.21 0.56 / 0.82 / 0.664 0.70 / 0.67 / 0.683 0.57 / 0.55 / 0.561 D2 0.17 0.59 / 0.89 / 0.708 0.86 / 0.83 / 0.844 0.64 / 0.63 / 0.637 D3 0.38 0.93 / 0.88 / 0.904 0.81 / 0.43 / 0.558 0.73 / 0.39 / 0.505 D4 0.14 0.30 / 0.64 / 0.408 0.64 / 0.56 / 0.598 0.44 / 0.39 / 0.412 Summarizing: C3 and C4 contain elks and deer that stay away from cattle (C3 moving in higher X than C4 ); C1 also contains only elks and deer, but those moves in the higher Y ex area where also the cattles move; C2 is the cattle cluster and co it contains also few elks and deer; ﬁnally C5 is another mixed be cluster which overlaps with C2 only for the cattle and28withMonday, September 26, 2011
- 40. overlapping TABLE IIIerlapping Comparing clusterings cost based on distance on the SCOP taxonomy, for 4] and the different values of p, the maximum number of labels per protein.gns all SCPS OCC-ISECT-p1 OCC-ISECT-p2 OCC-ISECT-p3 D1 0.231 0.196 0.194 0.193OCC-JACC D2 0.188 0.112 0.107 0.106call/F-score D3 0.215 0.214 0.214 0.231 .55 / 0.561 D4 0.289 0.139 0.133 0.139 .63 / 0.637 SCPS OCC-JACC-p1 OCC-JACC-p2 OCC-JACC-p3 .39 / 0.505 .39 / 0.412 D1 0.231 0.208 0.202 0.205 D2 0.188 0.137 0.130 0.127 D3 0.215 0.243 0.242 0.221that stay D4 0.289 0.158 0.141 0.152 C1 alsohigher Y extremely close in the taxonomy, the error should have a smalluster and cost. Following this intuition we deﬁne the SCOP similarityer mixed between two proteins as follows:and with d(lca(u, v)) sim(u, v) = , (8) max(d(u), d(v)) − 1 29 Monday, September 26, 2011
- 41. future work - scaling up - approximation algorithm - jaccard triangulation - more experimentation and applications 30Monday, September 26, 2011
- 42. thank you!Monday, September 26, 2011

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